Abstract:
In the short study, we present a contradiction which comes from ignoring a hidden assumption in Heine-Borel theorem. Finally, we prove a useful theorem which rectifies the contradiction. 2010 Mathematics subject Classification: 30L99 Keywords and Phrases: Metric Space, Compactness, Heine-Borel Theorem

Abstract:
In this work, we introduce the concepts of compactly invariant and uniformly invariant. Also we define sometimes C-invariant closed subspaces and then prove every m-dimensional normed space with m > 1 has a nontrivial sometimes C-invariant closed subspace. Sequentially C-invariant closed subspaces are also introduced. Next, An open problem on the connection between compactly invariant and uniformly invariant normed spaces has been posed. Finally, we prove a theorem on the existence of a positive operator on a strict uniformly invariant Hilbert space. DOI: http://dx.doi.org/10.3126/bibechana.v10i0.7555 BIBECHANA 10 (2014) 31-33

Abstract:
In this study, we obtain Banach algebras which norm of their unit elements is not one. These Banach algebras are subsets of Rk. Also, we present some interesting properties.

Abstract:
Let be an injective function. For a vertex labeling f, the induced edge labeling is defined by, or ; then, the edge labels are distinct and are from . Then f is called a root square mean labeling of G. In this paper, we prove root square mean labeling of some degree splitting graphs.

Abstract:
We investigate the FFT (Fast Fourier Transform) model and G-CSF (granulocyte colony-stimulating factor) treatment of CN (Cyclical Neutropenia). We collect grey collies and normal dog’s data from CN and analyze the G-CSF treatment. The model develops the dynamics of circulating blood cells before and after the G-CSF treatment. This is quite natural and useful for the collection of laboratory data for investigation. The proposed interventions are practical. This reduces the quantity of G-CSF required for potential maintenance. This model gives us good result in treatment. The changes would be practical and reduce the risk side as well as the cost of treatment in G-CSF.

Aminoguanidine
lanthanide thiodipropionate hydrates of composition [Ln(Agun)_{2}(tdp)_{3}·nH_{2}O], Agun = Aminoguanidine, tdp =
thiodipropionic acid, where Ln = La, Pr, Nd and Sm if n = 2, have been prepared and characterized by physic-chemical
techniques.

Abstract:
In the history of mathematics
different methods have been used to detect if a number is prime or not. In this
paper a new one will be shown. It will be demonstrated that if the following
equation is zero for a certain number p,
this number p would be prime. And
being m an integer number higher than (the lowest, the most efficient the operation). . If the result is an integer, this result will tell
us how many permutations of two divisors, the input number has. As you can
check, no recurrent division by odd or prime numbers is done, to check if the
number is prime or has divisors. To get to this point, we will do the
following. First, we will create a domain with all the composite numbers. This
is easy, as you can just multiply one by one all the integers (greater or equal
than 2) in that domain. So, you will get all the composite numbers (not getting
any prime) in that domain. Then, we will use the Fourier transform to change
from this original domain (called discrete time domain in this regards) to the
frequency domain. There, we can check, using Parseval’s theorem, if a certain
number is there or not. The use of Parseval’s theorem leads to the above
integral. If the number p that we
want to check is not in the domain, the result of the integral is zero and the
number is a prime. If instead, the result is an integer, this integer will tell
us how many permutations of two divisors the number p has. And, in consequence information how many factors, the number p has. So, for any number p lower than 2m？- 1, you can check if it is prime or not, just making the
numerical definite integration. We will apply this integral in a computer
program to check the efficiency of the operation. We will check, if no further
developments are done, the numerical integration is inefficient computing-wise
compared with brute-force checking. To be added, is the question regarding the
level of accuracy needed (number of decimals and number of steps in the
numerical integration) to have a reliable result for large numbers. This will
be commented on the paper, but a separate study will be needed to have detailed
conclusions. Of course,

Abstract:
Rain attenuation values were calculated using empirical raindrop-size distributions, which were, Marshall-Palmer (M-P), Best, Polyakova-Shifrin (P-S) and Weibull raindrop-size distributions, and also calculated using a specific rain attenuation model for prediction methods recommended by ITU-R. Measurements of Terahertz wave taken at 313 GHz (0.96 mm) were compared with our calculations. Results showed that the propagation experiment was in very good agreement with a calculation from the specific attenuation model for use in prediction methods by ITU-R.

Abstract:
If are the eigen values of a p-vertex graph , the energy of is . If , then is said to be hyperenergetic. We show that the Frucht graph, the graph used in the proof of well known Frucht’s theorem, is not hyperenergetic. Thus showing that every abstract group is isomorphic to the automorphism group of some non-hyperenergetic graph. AMS Mathematics Subject Classification: 05C50, 05C35